One of the best font is the “DejaVu” suite. “DejaVu Sans Mono” for fixed-width and
“DejaVu Sans” for variable-width.
This font also contains most of the math symbols in Unicode.
[see Math Symbols in Unicode]

Emacs using font DejaVu Sans Mono

GNU Unifont

GNU Unifont is a bitmapped font. (it's converted to scalable format (TrueType) in 2008 by Paul Hardy, but still bitmapped by design). This means, if you make the font size large, you'll see jagged edges.

GNU Unifont when enlarged.

Note: There's also Fixedsys and Code2000 that contains extensive Unicode characters (more than DejaVu). However, their quality is very bad.

Best Variable-width Font for Unicode

Two best proportional font for Unicode are DejaVu Sans and Arial Unicode MS. Both are sans-serif.

DejaVu Sans

DejaVu fonts is a suite of fonts containing both fixed-width (monospaced) and variable-width fonts.

Arial Unicode MS

On Microsoft Windows, you should try Arial Unicode MS. It comes with Microsoft Office, and is also in Mac OS X 10.5. (if you don't have it, you might be able to get it from Microsoft office free trial.)

Arial Unicode MS in emacs displaying Unicode chars.

Note that Arial Unicode MS has big gap between lines. So, you may not like it for programing. (but if you use emacs, you can adjust line spacing. [see Emacs: Set Line Spacing])

Font for Unicode Emoji 😸 👍

Which Font Has All Unicode Chars?

No single font has ALL Unicode chars. At best, they contain all chars in the
Basic Multilingual Plane (BMP).
The BMP contains the most common 65.536 k chars. (Unicode has 16 other “planes”, but these are rarely used.)

Font Substitution Tech

{Microsoft Windows, Mac OS X, Linux} all have Font substitution technology. When a char in the current font doesn't have a glyph, the system search for the glyph in other font set. So, this way, each font can be specially designed for a particular language or purpose, while you can still display many Unicode symbols in the same page.

StackOverflow answers: “How does a Unicode character get mapped to a glyph in a font?”: stackoverflow.com.